How to bevel selected edges in non-destructive way?
You can set bevel weights on individual edges.

To enable this feature in the modifier, select Weight



Now go into edit mode and make sure you are in edge selection



Now select the edges you want to bevel (shift click for multi-select).

Now go to the Edge menu and select Edge Bevel Weight



Move the mouse to modify the value.



Once done you can use the panel as shown:



You can fine tune this after by hitting F9 to modify last operator.

Update

So while this works for most geometries, it seems that something odd has happened in your file.  I suspect that one of your addons has already been weighting your edges and combined with what we are trying to do here, makes for an unhappy mesh.

So I removed all you modifiers and created a vertex group with 4 vertices for the excision ( cut-out). Then, I recreated your bevel modifier with 40mm. The trick to getting this working correctly is adding an edge-loop in the center as shown.



Now you should have your rough cutout shape.  Here's the settings I used for this first modifier.

N.B. you must use Width Method: Width here:



So now it's time to setup the second bevel modifier weights as described above.

Make sure to pay attention to which edges have weight here:



And finally the settings I have used for this second bevel modifier are:



And the final geometry: